GF(2.sup.n) denotes a Galois field containing 2.sup.n elements, wherein n is greater than 1. Said field is a number system in which there are 2.sup.n elements and in which the rules of addition and multiplication correspond to arithmetic modulo an irreducible polynomial of degree n with coefficients in G(2), G(2) being a number system in which the only elements are the binary numbers 0 and 1 and the rules of addition and multiplication are: 0+0=1+1=0; 0+1=1+0=1; 0.times.0=1.times.0=0.times.1=0; 1.times.1=1. The conventional approach to performing operations in GF(2.sup.n) involves choosing a polynomial P(x) of degree n which is irreducible over GF(2.sup.m), m&lt;n defining an element .alpha. in GF(2.sup.n) as a root of P(x)--satisfying P(.alpha.)=0--and assigning the unit vectors of length n with binary components to the elements 1, .alpha., .alpha..sup.2, . . . , .alpha..sup.n-1.
Exponentiation over GF(2.sup.n) is an operation that is required for many purposes, described in prior art literature, three of which--message authentication, user identification and exchange of keys--will be recalled here.